Fractional Bilinear Stochastic Equations with the Drift in the First Fractional Chaos

Abstract

In the paper we compute the explicit form of the fractional chaos decomposition of the solution of a fractional stochastic bilinear equation with the drift in the fractional chaos of order one and initial condition in a finite fractional chaos. The large deviations principle is also obtained for the one-dimensional distributions of the solution of the equation perturbed by a small noise.     

基于非线性梁理论的有限质点法

有限质点法是以向量式力学为基础,用有限数量的质点来模拟结构的变形行为,质点的运动由牛顿运动定律来计算。在有限质点法中,质点通过构件相连,构件约束着质点的运动,并且其内力由质点的运动变量来描述。基于向量式力学的基本思想和非线性梁理论,提出了一种新的有限质点法,该方法在共旋单元坐标系中描述梁的非线性变形。以空间梁系结构为例,推导了计算构件内力的非线性公式,并考虑了弯扭耦合变形。通过两个连续欧拉角的变换公式得到共旋坐标系的旋转矩阵。与传统的有限质点法相比,本文提出的方法避免了刚体虚转动分析。通过四个结构的数值求解,验证了本文方法在计算结构大变形响应时具有较高的精度。     

Two-dimensional model of elastically coupled molecular motors

A flashing ratchet model of a two-headed molecular motor in a two-dimensional potential is proposed to simulate the hand-over-hand motion of kinesins.Extensive Langevin simulations of the model are performed.We discuss the dependences of motion and efficiency on the model parameters,including the external force and the temperature.A good qualitative agreement with the expected behavior is observed.     

A fast iterative method to find the matrix geometric mean of two HPD matrices

The purpose of this research is to present a novel scheme based on a quick iterative scheme for calculating the matrix geometric mean of two Hermitian positive definite (HPD) matrices. To do this, an iterative scheme with global convergence is constructed for the sign function using a novel three‐step root‐solver. It is proved that the new scheme is convergent and shown to have global convergence behavior for this target, when square matrices having no pure imaginary eigenvalues. Next, the constructed scheme is used and extended through a well‐known identity for the calculation of the matrix geometric mean of two HPD matrices. Ultimately, several experiments are collected to show its usefulness.     

Markov processes conditioned on their location at large exponential times

Suppose that ${\left(X_t\right)}_{t\ge 0}$ is a one-dimensional Brownian motion with negative drift $?\mu$. It is possible to make sense of conditioning this process to be in the state 0 at an independent exponential random time and if we kill the conditioned process at the exponential time the resulting process is Markov. If we let the rate parameter of the random time go to 0, then the limit of the killed Markov process evolves like $X$ conditioned to hit 0, after which time it behaves as $X$ killed at the last time $X$ visits 0. Equivalently, the limit process has the dynamics of the killed “bang–bang” Brownian motion that evolves like Brownian motion with positive drift $+\mu$ when it is negative, like Brownian motion with negative drift $?\mu$ when it is positive, and is killed according to the local time spent at 0.An extension of this result holds in great generality for a Borel right process conditioned to be in some state $a$ at an exponential random time, at which time it is killed. Our proofs involve understanding the Campbell measures associated with local times, the use of excursion theory, and the development of a suitable analogue of the “bang–bang” construction for a general Markov process.As examples, we consider the special case when the transient Borel right process is a one-dimensional diffusion. Characterizing the limiting conditioned and killed process via its infinitesimal generator leads to an investigation of the $h$-transforms of transient one-dimensional diffusion processes that goes beyond what is known and is of independent interest.     

Brownian motion with drift on spaces with varying dimension

Many properties of Brownian motion on spaces with varying dimension (BMVD in abbreviation) have been explored in Chen and Lou (2018). In this paper, we study Brownian motion with drift on spaces with varying dimension (BMVD with drift in abbreviation). Such a process can be conveniently defined by a regular Dirichlet form that is not necessarily symmetric. Through the method of Duhamel’s principle, it is established in this paper that the transition density of BMVD with drift has the same type of two-sided Gaussian bounds as that for BMVD (without drift). As a corollary, we derive Green function estimate for BMVD with drift.     

Mean curvature flow with free boundary – Type 2 singularities

In this paper we give sufficient conditions that guarantee the mean curvature flow with free boundary on a pinched cylinder develops a Type 2 curvature singularity. We additionally prove that Type 0 singularities may only occur at infinity.     

Generation and Propagation of Optical Superoscillatory Vortex Arrays

Superoscillation is an intriguing wave phenomenon which enables subwavelength features propagating into far field and hence has potential applications in super‐resolution microscopy as well as particle trapping and manipulation. While previous demonstrations mostly concentrate on designing complicated nanostructures for generating uncontrollable superoscillatory functions, here a new technique which allows for creating polynomially shaped superoscillatory functions that contain phase singularity arrays is demonstrated both theoretically and experimentally. Such a technique is implemented in optical experiments for the first time and controllable superoscillatory lobes with feature much below the diffraction limit is achieved. More importantly, a general theoretical framework, which, to our knowledge, has not been reported before, is developed to show how the created superoscillations propagate to a distance of many Rayleigh ranges and eventually disappear when the distance is sufficiently larger. The validity of the model is confirmed by the experiments. The results may trigger further studies in light field shaping and manipulations in subwavelength scale.